Rigidity and Cohomology of Seaweed Lie Algebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908978408062976 |
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| author | Coll, Jr., Vincent E. Hylton, Alan |
| author_facet | Coll, Jr., Vincent E. Hylton, Alan |
| contents | Seaweed (biparabolic) subalgebras form a large and structurally rich class of subalgebras of simple Lie algebras. We determine their adjoint cohomology.
If $\mathfrak{s}$ is an indecomposable seaweed subalgebra of a complex simple Lie algebra, then \[ H^\ast(\mathfrak{s},\mathfrak{s})=0, \] and hence $\mathfrak{s}$ is absolutely rigid. If $\mathfrak{s}$ is decomposable, then the Coll--Gerstenhaber decomposition for Lie semidirect products gives, for each $n\ge 0$, a canonical description of $H^n(\mathfrak{s},\mathfrak{s})$ in terms of exterior powers of $\mathcal{Z}(\mathfrak{s})^\ast$ and the zero-weight cohomology of $\mathfrak{s}/\mathcal{Z}(\mathfrak{s})$. In particular, the center is the unique source of nontrivial adjoint cohomology.
These results identify indecomposability as the precise condition for cohomological rigidity and give a uniform description of adjoint cohomology for seaweed Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17675 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity and Cohomology of Seaweed Lie Algebras Coll, Jr., Vincent E. Hylton, Alan Rings and Algebras Seaweed (biparabolic) subalgebras form a large and structurally rich class of subalgebras of simple Lie algebras. We determine their adjoint cohomology. If $\mathfrak{s}$ is an indecomposable seaweed subalgebra of a complex simple Lie algebra, then \[ H^\ast(\mathfrak{s},\mathfrak{s})=0, \] and hence $\mathfrak{s}$ is absolutely rigid. If $\mathfrak{s}$ is decomposable, then the Coll--Gerstenhaber decomposition for Lie semidirect products gives, for each $n\ge 0$, a canonical description of $H^n(\mathfrak{s},\mathfrak{s})$ in terms of exterior powers of $\mathcal{Z}(\mathfrak{s})^\ast$ and the zero-weight cohomology of $\mathfrak{s}/\mathcal{Z}(\mathfrak{s})$. In particular, the center is the unique source of nontrivial adjoint cohomology. These results identify indecomposability as the precise condition for cohomological rigidity and give a uniform description of adjoint cohomology for seaweed Lie algebras. |
| title | Rigidity and Cohomology of Seaweed Lie Algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2604.17675 |